Questions tagged [inverse-function-theorem]
Recommended to be used when the Inverse function theorem is being employed in a question, and also for those users that need help understanding it.
380 questions
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Banach Implicit Function Theorem - Obtaining implicit derivatives of parameters while keeping functional equality fixed
Suppose I have a nonlinear real function $F(x, \omega, \xi)$, where $x\in\mathcal{X}, \omega\in\Omega, \xi\in\Xi$. Here, $\mathcal{X}, \Omega, \Xi$ are subsets of $\mathbb{R}^n, \mathbb{R}^m,\mathbb{R}...
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If $f:A\to\mathbb{R}^n$ is of class $C^1$ and $\det Df(a)\neq 0$ for $a\in A$, then $f$ is injective near the point $a$. Munkres Analysis on Manifolds
I am reading "Analysis on Manifolds" by James R. Munkres. Note that $|v|$ for $v\in\mathbb{R}^n$ and $|M|$ for a real matrix $M$ are sup norms in the following proof. The proof of Lemma 8.1 ...
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3 votes
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Problem about the differentiability of an inverse function
I want to prove the following Proposition. Let $V,W\subseteq\mathbb R^n$ be open, and let $$ F:V\to W $$ be a continuously differentiable bijection such that $F(0)=0$, $J_F(0)=I_n$, $\det J_F(u)\neq0$...
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Are the injectivity assumptions in different proofs of the Inverse function theorem necessary?
Let $A$ be an open set of $\mathbb{R}^n$ and let $f = (f_1,f_2,...,f_n):A \to \mathbb{R}^n$ a function of class $C^1(A)$. Let $x_0 \in A$ such that $\det(Jf(x_0)) \ne0$. Then, there exist an open ...
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we may take to be of the form $A\times B$, such that $F:A\times B\to W$ has a differentiable inverse $h:W\to A\times B$. Spivak Calculus on Manifolds.
I am reading "Calculus on Manifolds" by Michael Spivak. Spivak wrote: which we may take to be of the form $A\times B$, such that $F:A\times B\to W$ has a differentiable inverse $h:W\to A\...
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Problem 2-37. (a) in "Calculus on Manifolds" by Michael Spivak. Why did the author assume $D_1f(x,y)\neq 0$ for all $(x,y)$ in some open set $A$?
I am reading "Calculus on Manifolds" by Michael Spivak. 2-37. (a) Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuously differentiable function. Show that $f$ is not $1$-$1$. Hint: If, for ...
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f bijective, C1 and invertible differential at every point $\implies$ C1 inverse
Suppose we have a function function $f\colon U\to V$ where $U,V\subset \mathbb{R}^n$ are open subsets which is bijective, $\mathcal{C}^1$ and has invertible differential $Df(x)$ for all $x\in U$. ...
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Step in Theorem 5.2 (Inverse Function Theorem) of Serge's Fundamentals of Differential Geometry
In the proof below, why is $|g_y(x)|\le r$? The inequalities $|y|\le r/2$ and $|x|\le r$ seem to me only yield $$|g_y(x)| = |y + x - f(x)| \le \frac{r}{2} + r - |f(x)| \le \frac{3}{2}r.$$ Theorem 5.2:...
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